Toeplitz Operators On The Dirichlet Space

The operator theory on function spaces is one of the significant branch in func-tional analysis. In this dissertation, we mainly study the Toeplitz operators on the Dirichlet space and the Larger Dirichlet space and the Dirichlet space of unit polydisk, and focus on essential commutativity, self adjointness, normality, the compactness of finite sums of Toeplitz products and the compactness of Toeplitz operator.In chapter 1, we introduce some related backgrounds about the Dirichlet space and the Toeplitz operator on it, and give some basic definitions and symbols. Finally we introduce our research content.In chapter 2, we characterize the products of two Toeplitz operators on the Larger Dirichlet space of the unit disk, we prove that the products of two Toeplitz opera-tors Tf, Tg with the symbols of harmonic functions is compact perturbation of another Toeplitz operators Th if and only if the Berezin transform of fg-h is zero on (?)D.In chapter 3, we characterize the self adjointness and normality of Toeplitz op-erators with the symbol of W1,∞(its definition is given in chapter 3) on the Dirichlet space of the unit disk. We prove that Tu is self-adjoint if and only if u is a real constant function, solve the normality of Tu partly.In chapter 4, we characterize the compactness of finite sums of Toeplitz products on the Larger Dirichlet space of the unit disk. We prove that the finite sums of Toeplitz products is compact if and only if the finite sums of symbol functions is zero on (?)D. In chapter 5, we characterize the boundedness and compactness of Toeplitz oper-ator with the symbol of measure on the Dirichlet space of unit polydisk. We prove that the Toeplitz operators with bounded symols is compact.